Fig 1.
Schematic overview of the model.
The biomechanical state of the system during the n-th step is modeled by position, xn, and velocity, vn of a point mass connected to the ground via a rigid, massless leg of length l at point pn. The controller selects a foot placement location based on a proportional-derivative controller with the difference of the point mass from the contact point, xn − pn and its rate of change, vn, with gain factors bp and bd, as well as a constant offset, bo.
Fig 2.
Example solutions for the walking system with three different initial conditions.
The left column shows progressive walking in the anterior-posterior direction, the center column alternating walking in the medial-lateral direction, and the right panel combines both directions to show a top-down view of the path. Panels A and B show the absolute position of the CoM, x(t), (solid) and CoP, p(t), (dotted) vs. time, Panels C and D show the CoM velocity, v, and Panels E and F show the position of the CoM relative to the CoP, q(t) = x(t) − p(t). The initial conditions were chosen slightly different, but the system parameters were the same in all three cases.
Fig 3.
Phase plot of the periodic orbits for progressive (Panel A) and alternating stepping (Panel B).
The horizontal axis is position of the CoM, the vertical axis is velocity. The thick blue line is the orbit. Here, the solutions are shown over 1 s, starting at midstance and including two steps. Gray arrows in the blue orbit indicate direction and time, spaced 0.05 s apart. The thin gray curves are other orbits. The vertical gray lines indicate an instantaneous change in the dynamics from taking a step, and the orbit has a cusp at these points. The contact point, pn, is shown in orange for each step.
Fig 4.
Examples of solutions with four different values of the derivative gain parameter, bd.
Panel A shows the CoM position, Panel B the CoM velocity, vs. time. The initial condition is the same in all four systems, and the solution over the first second is also identical (gray). At t = 1 s, a small perturbation is applied, which changes the solution of the system depending on the value of the control gain, bd.
Fig 5.
Stability region in the parameter space spanned by bp and bp.
White indicates that the system is unstable. The colored triangle is the region of stable walking systems, with color corresponding to the largest absolute eigenvalue, ρ, of the system’s transition matrix A.
Fig 6.
The limits of the stability region change with cadence.
Panel A shows the triangular stability region for two example cadences, 80 (blue) and 110 (yellow) steps per minute. The dots show estimated parameter values from humans walking to metronomes at that cadence. The right side shows how the limits of the stability region change with cadence. The gray areas represent the projections of the stability region onto the bp (Panel B) and the bd axes (Panel C), vs. cadence on the horizontal. The dots are parameter estimates from the same human data as in Panel A, plotted here against the average cadence each participant walked at The cadence for each participant is the average for that trial, which usually differs by some degree from the paces imposed by the metronome.
Fig 7.
Gain parameter values estimated from human data change when restricting the ankle roll mechanism.
Bars show the average proportional (Panel A) and derivative (Panel B) gain parameter estimate from N = 30 participants, with normal walking on the left and ankle roll restricted by a ridge attached to the sole of the shoes on the right. Gray dots represent individual participants, with gray lines connecting data from the same participant in both conditions.